What does it mean to be good at math? This is something I’ve been musing over for some time now.

I used to think that someone who was good at math was someone who could solve many different types of problems. By hand. The long way. The way I was taught to do it.

I’m not too sure anymore. There are tools available that help us solve equations. That let us solve by graphing. That let us solve with CAS. Why isn’t it okay to use these to get beyond the symbolic manipulation that often frustrates students? Why isn’t it okay to use the tools that let them get to the real problem solving?

If a student can read a word problem, understand what it is asking, set up equations to model the situation, use a graphing calculator to get the solutions, and evaluate the reasonableness of the solution why isn’t that enough? For the “average” student, does it matter if they say the answers are or if they say approximately 1.412 and -0.079?

If a student can solve a problem by hand and get the answer of is that not enough? Why is a better answer?

I want my students to understand the relationship between the graph, the table, the equation, and the situation. I want my students to be able to explain and defend their answers. I want them to be able to evaluate the reasonableness of the solutions proposed by others.

There are tools that let more students have access to these ideas. Why do we insist they do it by hand?

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16 Responses to Using the tools

I like that you’re considering these issues. There is certainly a time and place to use computational tools to help speed the problem-solving process. There are also a couple good reasons I can think of to require long-form work:

1. Algebraic manipulation practice. (Factoring and simplifying are needed in many other mathematical contexts, so it makes sense to ask students to do them at every opportunity.)
2. Multitasking. (What I mean is: Diving into a sub-part of a task and then coming back to the original task, while recording your work so you can re-trace your steps, is a skill that requires practice.)

Your big-picture goals for your students are the right ones. But I don’t think you’ll get them there without giving them practice doing the work.

This: “Your big-picture goals for your students are the right ones. But I don’t think you’ll get them there without giving them practice doing the work.”

To me, it would appear that the reasons for teaching something that can be done better another way is to avoid the “other way” (computer, CAS, slide rule, whatever) becoming a black box. Once it’s a black box, it becomes hard to use constructively. Your calculator can solve for x. Great. Can it solve for y? How do you know? What happens if you want a different form of the answer it gives? What if it only gives a decimal answer and you need to manipulate it [exactly] further later? Also, “magic happens here” doesn’t go over too well in most proofs.

So, practice in the smaller details is good. We’re writing applications that deal with memory handling in early CS courses, in order to make our own lists. Perfectly good, standard, well-implemented lists already exist, with dozens more features than we’ll ever give them, but we’re not using them yet, so we can understand how they work when we use them later.

That said, there’s a line between “practice” and “mandating forever.” Proving you know something now and then is probably useful, but doing it every time probably isn’t. For example, my current math teacher will take any form of the right answer, as long as it’s right. (He claims.) At some point, there’s an assumption that you know how to do those things. They’ll still be necessary when you need to derive something or whatever (nicely, anyway), but there’s no reason to require proving it every answer.

That’s not saying you have to completely understand every tool you use. But it helps understand what you’re doing. Especially when you’re solving problems, which would seem to get back to your point.

Don’t learning goals depend on the student? I think that your emphasis on understanding how math can model situations with multiple representations is undeniably important for all students. And I think for most kids it really can stop there, when we think about how they will be using this learning to function in their day to day lives.

But for our future scientists, mathematicians, and engineers, I think we owe them some greater dexterity working with numbers and algebraic expressions. The trouble in many places being that these “types” of kids are all mixed up in our classes, and that it can be impossible to tell which are going to go in what direction. I tend to insist on exact values and restrict calculator use in my honors classes, and rely very heavily on the calculator and computer simulations in remedial classes. It’s the mixed ability regular track (we call them “regents”) classes that present the biggest challenge for where to aim.

I guess I should have clarified a bit more. When I was writing this I was thinking of the student for whom Algebra is a struggle. The student who sees the algebraic manipulation and shuts down. With the push to put everyone in Algebra, whether they are ready or not, why not use the tools that can help them?

I guess there is a part of me that feels the by hand skills are necessary – if one is going into a career that requires it. As everyone has said above, the factoring and algebraic manipulation skills are needed for college (although I’m still not sure why. Do Engineers really do this stuff by hand? Do Actuaries?).

Kate, your comment about the mixed ability classes are the ones of which I’m thinking too. If a student can experience success by solving graphically while the student sitting next to them can get the same result by hand and each student is challenged appropriately by each activity, isn’t that success?

Thanks for helping me think through this Ben, Andy, and Kate. Not that I’m fully convinced. 🙂

I have a couple thoughts about this. First, I don’t think that (the square root of 50) is more or less simple than 5*(the square root of 2) — I see them as equivalent, really, since you’ve got a square root either way. I run into this sometimes in Calculus when students feel like it’s better to write (x^2+3x+2) instead of (x+1)(x+2), whereas the truth is that sometimes it’s better to multiply the answer out and sometimes it’s better to factor it — it all depends on what you’re going to do next. As you can tell, I don’t spend a lot of time emphasizing writing the answer in a particular way.

That said, I do see the value of algebraic manipulation in some form. In your comment above about what careers use algebraic manipulation, what came to mind is statistics. Some of my students see the two formulas as completely different:

z=(Margin of Error)/(standard deviation of the mean)
(Margin of Error) = z*(standard deviation of the mean)

That’s simplifying it a little bit, since the standard deviation of the mean has a formula itself, but I do think that understanding statistics, say, is easier if you realize that all of the main formulas we use are just variations of one another, not individual bolts of lightening. And that takes understanding the relationships between the variables, which is what I see as a main idea in Algebra.

Hi,
I’m an engineer that’s been in industry for ~12 years now, and in the past I’ve taken a very judgmental approach to using graphing calculators, etc. in high school, thinking it’s a lazy way to learn. Then one of my colleagues (who was tired of my soap boxing) asked me if I used to compute sines/cosines by hand in high school, or if I used the “lazy” way of punching it into the calculator. Of course, I had used a calculator in high school and had to stop and think, where do you draw the line and what is the real value of doing things by hand.

So here’s a process and product engineering perspective. I use highly automated tools that automatically graph and curve fit input variables to output variables, whenever I just need to characterize an without a deep level of understanding of mechanisms or processes, just inputs and outputs. This is what I would call “turn the crank” engineering where there is a procedure for doing this – almost any engineer can do it.

BUT, the really big value comes when you can sit down and list the variables in your entity, assemble them from physics based principles, and prove to yourself and others that you can make predictions since you have the fundamental equations. This happens in engineerning, and when it does, it’s extraordinarily powerful. Now you have something that you can use over and over, and keep extending rather than starting over with every experiment.

The major point here is that it is important for students to recognize the levels of approach, and what they yield. Highly automated problem solvers very rarely yield highly inventive, unique, and powerful solutions. It’s the folks that can work the equations by hand and get a real feel for the mechanisms that will excel.

Thanks for sharing your perspective. I agree that understanding what the variables represent and their relationships is important. However, I don’t think that using a graphing calculator automatically precludes this from happening. If the good activities are developed and good questions are asked of the students.

I guess I’m still struggling with how to make Algebra accessible to everyone (and not having them detest problem solving) and push each student to reach their fullest potential.

Hmm, I try to help you how to teach counting the root of 676 to your students. Under number 6, which has roots is 4, the value of its root is 2. Save this 2 as a number of leading. Then multiply that number 2 with two (this tips from Mr. Calandra from India, who has found the root calculation method in the years around 1491), that we know it gives 4. After that, please ask them 4(x) times (x) is equal 276 (the difference between 676 and 400, because there are 4 in hundreds). Then save the x number behind the number 2 was leading, so that the root of 676 is 2(x). Of course teachers with easy call x is 6, because he already knows that 46 × 6 = 276. So they will say to his students that the root of 676 is 26, please check again would you 26 × 26 = 676 is not. But how about the students? Maybe they say that this method violate the rights of children to play.” Because the instinct of students will calculate how variations ten times to get the 276, ranging from 4(0)*(0), 4(1)*(1), etc…Indeed very tiring. How about you?

Maybe you all will agree with me that the Calandra method of finding the root of positive number has been popular method in the world. But apparently in the above example, the method still need great effort in yeilding the root of arbitrary number. For example in calculating the root of 26, even to the root of 4 if students don’t know that 4=2*2. According me the Calandra method is still difficult to be applied in the classroom. Whether all of you need an explicit formula to the root calculation? If so, please give me an information about appropriate journal to publish the root formula. Thx.